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Henry F. Mollet Moss Landing Marine Labs

Comparison of Elasticity Patterns of Elasmobranchs and Mammals with Review of Vital Rates of Lamnids. Henry F. Mollet Moss Landing Marine Labs. 1984 vs. 2001 Compagno FAO Catalogue, Lamnids Only.

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Henry F. Mollet Moss Landing Marine Labs

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  1. Comparison of Elasticity Patterns of Elasmobranchs and Mammals with Review of Vital Rates of Lamnids Henry F. Mollet Moss Landing Marine Labs

  2. 1984 vs. 2001 Compagno FAO Catalogue, Lamnids Only • Porbeagle:Much new info on reproduction (Francis and Stevens 2000; Natanson et al. 2002; Jenson et al. 2002; Campana et al. 2002) • Salmon shark:Much new info in press (Ken Goldman pers. comm. in Compagno 2001) • Longfin mako:Litter size 8 (Casey 1986 Abstract) • Shortfin mako:Litter size as large as 25-30 (Mollet et al. 2002); Gestation period 18 months and reproductive cycle probably 3 years (Mollet et al 2000); Age-at-maturity ~14 yr rather than ~7 yr (1 growth band pair/year, Campana et al. 2002 based on bomb C14-analysis of 1 vertebrae from 1 specimen) • White shark:Much new info on reproduction (Francis 1996; Uchida et al. 1996; both in Klimley and Ainley eds.). Gestation period 18? months with 3? yr repro cycle; length/age-at-maturity 5?m/ 15? yr (Mollet et al. 2000; Bruce et al. (in press). Crucial to get info on litters with early-term embryos.

  3. Sanzo (1912) and Uchida (1989) Shortfin Mako Embryos

  4. From Capture and ‘Recapture’ to Publications 90 years apart, 100 years to get correct ID

  5. ?White Shark Litter of ~5, ~ 30 cm TLTaiwan ~ 10 years ago (Victor Lin p.c.)

  6. Summary • Elasticities patterns for ALL Elasmos are nearly the same and can be done without a calculator, at the Rio Negro Beach. • Can predict E-pattern for Elasmos with little or no data.

  7. Background I • Elasticities give proportional changes of population growth () due to proportional change of vital rates (a): E(a) = dln()/dln(a) = (a/) ( d/da) • Elasticities are robust, don’t need accurate  • Vital rates of elasmos, in particular lamnids, are poorly known. Therefore precise population growth rates () are difficult to obtain anyway.

  8. Background II • Mollet and Cailliet (2003) reply to Miller et al. (2003): LHT or Leslie matrix are easier and safer than stage-based models. • Here, I’ll cover elasticity patterns, not just for one species, but for all elasmos and all mammals using data for 60 elasmos and 50 mammals.

  9. LHT and corresponding Leslie Matrices for a Hypothetical Species = 3 yr, m = 3, =5, P-juv = 0.631, P-adu = 1.0! (but m6 = 0)

  10. Elasticity Matrix and Elasticity Pattern by Summing Elasticities over Age-Classes

  11. , , and 3 Generation Times

  12. Elasticity pattern from  and Abar (Gestation period GP provides refinement) • E(fertility) = E(m) • E(juvenile survival) = E(js) • E(adult survival) = E(as) • 1/E(m) = <w,v> = Abar ! Dynamite!(<w,v> with w1 = 1, v1 = 1) • E(js) = ( - GP)/Abar • E(as) = (Abar -  + GP)/Abar • Normalization is easy

  13. Why Abar? The Short Version!

  14. The Crux of the Matter • E-pattern can be calculated from = age-at-first-reproduction,Abar= mean age of reproducing females at stable age distribution (=  x -x lx mx = f (vital parameters and )), GP = gestation period provides refinement. • Presents great simplification and allows better understanding of E-patterns even if we have to solve the characteristic equation to get Abar. • Elasticity matrix no longer needed for age-structured species.

  15. A Potential Problem and Proposed Solution for Elasmos • Catch 22 situation if we’d like to estimate the E-pattern of an Elasmo at the Rio Negro Beach? • If Abar/alpha were roughly constant, we could estimate Abar from the mean ratio and the elasticity pattern could be easily estimated without the need to solve the characteristic equation. • Example:  = 7 yr, Abar/ ~ 1.3, thus Abar ~ 9 yr:E(m) normalized = 1/(Abar + 1) = 1/10 = 10%E(juvenile survival) =  * E(m) = 7 * 10% = 70%E(adult survival) = 20% (from sum = 100%)

  16. (Abar/) Ratios for Elasmos and Mammals • Mean (Abar/) of 60 Elasmos:1.31, CV = 9.3%, Range 1.1 (S. lewini, S. canicula) - 1.8 (C. taurus); • Cortes (2002) Stochastic Calculation for n = 41 Elasmos, mean (Abar/) = 1.46, CV = 14.2%, range 1.1-2.0; • Mean (Abar/ ) of the 50 Mammals in Heppell et al. (2000): 2.44, CV = 33.5%, Range 1.2 (Snowshoe Hare) - 5.0 (Thar, l. Brown Bat)

  17. Normalized Elasticity Pattern for Elasmos using mean (Abar/) Ratio of 60 Elasmos

  18. Area Plot of Elasticity Pattern for Elasmos using Mean (Abar/) Ratio of 60 Elasmos

  19. Theory and E-patterns of Elasmos from LHT are in good agreement

  20. Shortfin and Longfin Mako Elasticity Patterns

  21. Salmon shark, Porbeagle, and White Shark Elasticity Patterns

  22. Elasticity Patterns of 4 RaysDasyatis violacea, Narcine entemedor, Myliobatis californica, Dipturus laevis

  23. Elasmos with Extreme Elasticity Patterns?Scyliorhinus canicula (m =105/2) , Sphyrna lewini,(m = 26-35/2) Carcharias taurus (m = 2/2x2), Rhincodon typus (m = 300/2x2?)

  24. Potential Problems • I used constant and fairly large mortalities for Elasmos. • They could be more Mammal like, larger mortality for juveniles and smaller mortality for adults. • Cortes (2002) used variable mortalities and Abar/ was still close to 1 and had low variability. • Elasticities only applicable to stable age distribution. True but E-patterns are very robust. Would have to move far from stable age distribution for E-pattern to become unsuitable for making management proposals.

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